for the three cubes shown below, determine their surface area, volume and surface area to volume ratio. Then circle the one you believe would be the most efficient and write a summary stating why.
1.5 cm x 1.5 cm x 1.5 cm
.5 cm x .5 cm x 6 cm
3 cm x 2 cm x 2 cm
If you compute volume <em>vs</em> surface area, you will find that the shape with highest V/A ratio is a cube.
To determine the surface area of a cube, you need to calculate the total area of all its faces. Since a cube has six equal sides, the formula for surface area is:
Surface Area = 6 * (length of a side)^2
To find the volume of a cube, you need to multiply the length of one side by itself twice (cube it), since all sides are equal. The formula for volume is:
Volume = (length of a side)^3
Now let's calculate the surface area, volume, and surface area to volume ratio for each of the three cubes:
1. Cube with sides of 1.5 cm x 1.5 cm x 1.5 cm:
- Surface Area: 6 * (1.5 cm)^2 = 6 * 2.25 cm^2 = 13.5 cm^2
- Volume: (1.5 cm)^3 = 3.375 cm^3
- Surface Area to Volume Ratio: 13.5 cm^2 / 3.375 cm^3 = 4 cm^-1
2. Cube with sides of 0.5 cm x 0.5 cm x 6 cm:
- Surface Area: 6 * (0.5 cm)^2 = 6 * 0.25 cm^2 = 1.5 cm^2
- Volume: (0.5 cm)^3 = 0.125 cm^3
- Surface Area to Volume Ratio: 1.5 cm^2 / 0.125 cm^3 = 12 cm^-1
3. Cube with sides of 3 cm x 2 cm x 2 cm:
- Surface Area: 6 * (3 cm)^2 = 6 * 9 cm^2 = 54 cm^2
- Volume: (3 cm)^3 = 27 cm^3
- Surface Area to Volume Ratio: 54 cm^2 / 27 cm^3 = 2 cm^-1
Now, let's evaluate the efficiency of each cube. The cube with the most efficient surface area to volume ratio is the one with the lowest value for this ratio. Comparing the values calculated above:
1. The cube with sides of 1.5 cm x 1.5 cm x 1.5 cm has a surface area to volume ratio of 4 cm^-1.
2. The cube with sides of 0.5 cm x 0.5 cm x 6 cm has a surface area to volume ratio of 12 cm^-1.
3. The cube with sides of 3 cm x 2 cm x 2 cm has a surface area to volume ratio of 2 cm^-1.
Therefore, the cube with sides of 3 cm x 2 cm x 2 cm has the most efficient surface area to volume ratio. This means that this cube has a relatively larger volume compared to its surface area, which can be advantageous in terms of material usage or insulation properties.
In summary, the cube with sides of 3 cm x 2 cm x 2 cm is the most efficient due to its lower surface area to volume ratio, indicating a better utilization of its volume in relation to its surface area.